EXTREMISATION OF JARLSKOG INVARIANTS

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EXTREMISATION OF JARLSKOG INVARIANTS
W. G. SCOTT RAL/SOTON MEET 3/3/06
P. F. Harrison and W. G. Scott
Phys. Lett. B 628 (2005) 93. hep-ph/0508012
WEAK-BASIS INV.
JARLSKOG INVARIANCE
e.g. for the quarks
Universal Weak Interact.
Universal Weak Interact.
U(3)
Diagonal Non-Diagonal
Non-Diagonal Diagonal
OBSERVABLES JARLKOG INVARIANT
FUNDAMENTAL LAWS JARLSKOG COVARIANT !!
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(WEAK-BASIS)
IN THE STANDARD MODEL
Up Mass Matrix
Down Mass Matrix
Universal Weak-Interaction
You can have any 2 but NOT all 3 matrices
diagonal!!
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THE ARCHITYPAL JARLSKOG INVARIANT
THE JARLSKOG DETERMINANT
The Determinant of the Commutator of mass
matrices
Extremising the Jarlskog Invariant J leads to

i.e. LEADS TO TRIMAXIMAL MIXING!!
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TRIMAXIMAL MIXING
Originally proposed for the quarks!!

HS PLB 333 (1994) 471. hep-ph/9406351
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TRI-BIMAX (HPS) MIXING

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S3 GROUP MIXING
MAGIC-SQUARE MIXING
(GENERALISES TRIMAX. AND TRI-BIMAX
MIXING)

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S3 GROUP MIXING
Magic-Square Mixing

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TRI-BIMAXIMAL (HPS) MIXING
(MINOS SOON!)
AT LEAST APPROXIMATELY !!!!
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THE 5/9-1/3-5/9 BATHTUB
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UP-TO-DATE FITS
A. Strumia and F. Vissani
Nucl.Phys. B726 (2005) 294.
hep-ph/0503246
IS THE BEST MEASURED MIXING ANGLE !!!
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FLAVOUR-SYMMETRIC
JARLSKOG INVARIANT MASS PARAMETERS
Charged-Leptons Mass Matrix
Neutrinos Mass
Matrix

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THE CHARACTERISTIC EQUATION
e.g. For the Charged-Lepton Masses

where
The Disciminant

ALL JARLSKOG INVARIANT!!
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EXTREMISATION A TRIVIAL EXAMPLE
In the SM
Yukawa couplings
Add to SM Action, the determinant
(taken here to be dimensionless) i. e.
e.g.
NOT BAD!!

HS PLB 333 (1994) 471. hep-ph/9406351
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MATRIX CALCULUS THEOREM
A any constant matrix, X a variable
matrix
WHEREBY e.g


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EXTREMISING Tr
(FOR FIXED MASSES)
With No Constraints
Differentiate Mass Constraints
Lagrange Multipliers
With Mass Constraints Implemented
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EXTREMISING Tr
(CONTINUED)
Eq. 1, off-diagonal elements, Re parts
Non-Trivial Solution
MAGIC-SQUARE CONSTRAINT!!
i.e.
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EXTREMISING Tr
(CONTINUED 2)
Eq.1 off-diagonal elements, Im parts
Non-Trivial Solution
CIRCULANT MASS-MATRIX i.e. TRIMAXIMAL MIXING!!!
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Increibly, all the remaining equations are
either redundant or serve only to fix the
lagrange multipliers
JARLSKOG SCALARS!!
Above remains true in all the extremisations we
performed!!
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THE SUM OF THE 2 x 2 PRINCIPAL MINOIRS
The K-matrix is the CP-symmetric analogue of
Jarlskog J
Plaquette Products
K-matrix
Extremise (in a hierachical approximation) wrt
PDG
2 x 2 MAX-MIX. ???

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SO NOW TRY EXTREMISING Tr
Eq. 1, off-diagonal elements, Re parts
Eq.1 off-diagonal elements, Im parts
Triv. Solns
2 x 2 MAX. MIX. !!
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2 x 2 MAXIMAL MIXING

Not Bad!! - but trivial 2 x 2 Max. solution
excluded by solar data!!

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EXTREMISING Tr
(CONTINUED)
Non-Trivial Solution (it
turns out, we need only consider
)
with
adjusted to give observed
Absolute masses not yet measured, but with the
minimalist assumption of a normal classic
fermionic neutrino spectrum
we have a unique prediction for the mixing
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NON-TRIVIAL CP-CONSERVING MIXING
Setting
SUGGESTIVE, BUT NOT CONSISTENT WITH DATA !!
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THE ASSOCIATED LAGRANGE MULTIPLIERS
Fixing the Lagrange multipliers
Assume the Non-Trivial Solution
These Lagrange Mults. are specific to the
non-trivial soln.
i.e. they fail for the 2 x 2 Max. solution!!!
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A COMPLETE SET OF MIXING VARIABLES
Higher powers of L,N need not be considered by
virtue of the characteristic equation hence
9 Quadratic Commutator Invariants, of which 4
are functionally independent, e.g.
(flavour-symmetric mixing variables!)
The Q-matrix is a moment-transform of the
K-matrix
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EXTREMISE IMPROVED EFFECTIVE ACTION
,AntiCommutator
Gives trajectory of solutions depending on the
parameter q To locate realistic soln. impose
magic-square constraint
n.b. The inherent cyclic symmetry of the solution
means that the magic-square constraint removes
one parameter - not two.
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NON-TRIVIAL CP-CONSERVING MIXING
Focus on pole at
and deviations
Setting
COVARIANT STATEMENT OF REALISTIC MIXING!!!
i.e. APPROX. HPS MIXING !!!
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And finally, the associated Lagrange
Multipliers
Where eg.
When we have the perfect action all LMs will
vanish!!
KOIDES RELATION

Y. Koide, Lett. Nuov. Cim. 34 (1982) 201.
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CONCLUSIONS
1) Extremise Tr C3 -gt tri-max
2) Extremise Tr C2 -gt 2 x 2-max
non-trivial solution not in agreement with
experiment

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SPARE SLIDES

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SYMMETRIES OF HPS MIXING
e.g.
M 0 SUBSET OF CLEBSCH- GORDAN COEFFS.
COULD PERHAPS BE A USEFUL REMARK ?!!
See J. D. Bjorken, P. F. Harrison and W.G.
Scott. hep-ph/0511201
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TRI-BIMAXIMAL (HPS) MIXING
ATMOS.
AT LEAST APPROXIMATELY !!!!
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W. G. SCOTT _at_ RL . AC . UK CERN-TH-SEMINAR
13/01/06
A VARIATIONAL PRINCIPLE IN ACTION?
P. F. Harrison, D. H. Perkins and W. G. Scott
Phys. Lett. B 530 (2002) 167. hep-ph/0202074
TRI-BIMAXIMAL (HPS)-MIXING
P. F. Harrison and W. G. Scott
Phys. Lett. B 535 (2002) 163.
hep-ph/0203209 Phys. Lett. B 547 (2002) 219.
hep-ph/0219197 Phys. Lett. B 557 (2003) 76.
hep-ph/0302025 Phys. Lett. B 594 (2004) 324.
hep-ph/0403278
SYMMETRIES DEMOCRACY MUTAUTIVITY

EXTREMISATION
Phys. Lett. B 628 (2005) 93. hep-ph/0508012
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TRI-BIMAXIMAL (HPS) MIXING
IS PHASE- CONVENTION INDEPENDENT
AT LEAST APPROXIMATELY !!!!
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TRIBIMAXIMAL (HPS) MIXING
HPS PLB 458 (1999) 79. hep-ph/9904297 WGS
hep-ph/0010335
c.f. G. Altarelli and F. Feruglio hep-ph/980735
3 with
AT LEAST APPROXIMATELY !!!!
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TRI-BIMAXIMAL (HPS) MIXING
ATMOS.
AT LEAST APPROXIMATELY !!!!
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Oscillation 37.8/40
Decay
49.2/40 Decoherence 52.4/40
M. Ishituka hep-ph/0406076

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TRI-BIMAXIMAL (HPS) MIXING
REACT.
ATMOS.
AT LEAST APPROXIMATELY !!!!
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TRIMAXIMAL MIXING)

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T. Araki et al. hep-ex/0406035
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TRI-BIMAXIMAL (HPS) MIXING
REACT.
ATMOS.
AT LEAST APPROXIMATELY !!!!
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TRI-BIMAXIMAL (HPS) MIXING
SOLAR
ATMOS.
AT LEAST APPROXIMATELY !!!!
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TRI-BIMAXIMAL (HPS) MIXING
AT LEAST APPROXIMATELY !!!!
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TRI-BIMAXIMAL (HPS) MIXING
SOLAR
ATMOS.
AT LEAST APPROXIMATELY !!!!
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TRIMAXIMAL MIXING
HS PLB 333 (1994) 471. hep-ph/9406351 (for
the quarks!) HPS PLB 349 (1995) 357.
http//hepunx.rl.ac.uk/scottw/ L. Wolfenstein
PRD 18 (1978) 958. N. Cabibbo PL 72B (1978) 222.
MAXIMAL CP-VIOLATION !!
(cf. C3 CHARACTER TABLE)
N. Cabibbo
We are probably far from this. . but not
very far
Lepton-Photon 2001
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MASS MATRICES
(ASSUMED HERMITIAN
3 x 3 circulant
2 x 2 circulant
Diagonalise
eigen-vecs eigen-vals
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S3 GROUP MATRIX
(FLAVOUR BASIS)
(i.e. charged-leptons diagonal)
NAT. REP.
RETRO-CIRC.
CIRC.
S3 GROUP MIXING
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S3 GROUP MIXING
(TRI-
MAX. MIXING)
GENERALISES TBM

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S3 GROUP MIXING
(TRI-
MAX. MIXING)
GENERALISES TBM
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c.f. The Democratic Mass matrix
An S3 GROUP MATRX Commutes with
THE DEMOCRACY OPERATOR
(and the converse)
(S3 CLASS OPERATOR)
DENICRACY SYMMETRY/INVARIANCE
Conserved Quantum Nos. etc.
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SO FINALLY
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TRI-MAXIMAL MIXING
HPS PLB 349 (1995) 357 N. Cabibbo PL 72B (1978)
222.
(cf. C3 CHARACTER TABLE)
N. Cabibbo
We are probably far from this. . but not
very far
Lepton-Photon 2001
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TRI-BIMAXIMAL (HPS) MIXING
ROWS/COLUMNS SUM TO UNITY
AT LEAST APPROXIMATELY !!!!
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FLAVOUR-SYMMETRIC MIXING INVARIANTS







1) The Determinant of the Commutator
ie. TRIMAX. MIX!!
2) The Sum of the 2x2 Principal Minors
K-matrix

TRI-BIMAX ???
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